
138 MATHEMATICS
Observe that as the speed increases, time taken to cover the same distance decreases.
As Zaheeda doubles her speed by running, time
reduces to half. As she increases her speed to three
times by cycling, time decreases to one third.
Similarly, as she increases her speed to 15 times,
time decreases to one fifteenth. (Or, in other words
the ratio by which time decreases is inverse of the
ratio by which the corresponding speed increases).
Can we say that speed and time change inversely
in proportion?
Let us consider another example. A school wants to spend ' 6000 on mathematics
textbooks. How many books could be bought at ' 40 each? Clearly 150 books can be
bought. If the price of a textbook is more than ' 40, then the number of books which
could be purchased with the same amount of money would be less than 150. Observe the
following table.
Price of each book (in ') 40 50
60 75 80
100
Number of books that 150 120 100 80 75 60
can be bought
What do you observe? You will appreciate that as the price of the books increases,
the number of books that can be bought, keeping the fund constant, will decrease.
Ratio by which the price of books increases when going from 40 to 50 is 4 : 5, and the
ratio by which the corresponding number of books decreases from 150 to 120 is 5 : 4.
This means that the two ratios are inverses of each other.
Notice that the product of the corresponding values of the two quantities is constant;
that is, 40 × 150 = 50 × 120 = 6000.
If we represent the price of one book as x and the number of books bought as y, then
as x increases y decreases and vice-versa. It is important to note that the product xy
remains constant. We say that x varies inversely with y and y varies inversely with x. Thus
two quantities x and y are said to vary in inverse proportion, if there exists a relation
of the type xy = k between them, k being a constant. If y
1
, y
2
are the values of y
corresponding to the values x
1
, x
2
of x respectively then x
1
y
1
= x
2
y
2
(= k), or
.
We say that x and y are in inverse proportion.
Hence, in this example, cost of a book and number of books purchased in a fixed
amount are inversely proportional. Similarly, speed of a vehicle and the time taken to
cover a fixed distance changes in inverse proportion.
Think of more such examples of pairs of quantities that vary in inverse proportion. You
may now have a look at the furniture – arranging problem, stated in the introductory part
of this chapter.
Here is an activity for better understanding of the inverse proportion.
Multiplicative inverse of a number
is its reciprocal. Thus,
is the
inverse of 2 and vice versa. (Note
that
).